Some precision manufacturing technologies use mathematical models of complex surfaces to represent manufacturable objects. Such models include Catmull-Clark surfaces and non-uniform rational B-spline (NURBS) surfaces. Such a model takes the form of a polyhedral mesh that is a level of approximation to a continuous representation of a manufacturable object. The model may be refined from a relatively coarse mesh to a finer mesh in a recursive fashion until a level of smoothness has been achieved. Conventional approaches to refining a model of a complex surface include defining refinement rules for the control points of a patch region of a mesh: a vertex, edge points that lie on edges of the mesh connecting control points, and face points that lie in the center of faces of the mesh.